Tagged Questions

26 views

49 views

810 views

Taylor's Theorem for Multivariate Functions

Please look at this theorem in Wiki regarding Taylor's theorem generalized to multivariate functions: Multivariate version of Taylor's Theorem The version stated there is one that I'm not familiar ...
70 views

Taylor expansion in $4D$

Let $f(x)=(x_2,-x_1,\sqrt 2 x_4 + x_1^3,-\sqrt 2x_3+x_3x_4^2)$ be a vector valued function from $\mathbb R^4\to\mathbb R^4$. Would anyone help me expand it up to and including the third term in its ...
257 views

Determine whether a multi-variable limit exists

I need to determine whether the next limit exists: $$\lim_{(x,y)\to(0,0)}f(x,y)=\lim_{(x,y)\to(0,0)}\frac{\cos x-1-\frac{x^2}2}{x^4+y^4}$$ Looking at the numerator $(-1-\frac{x^2}2)$ it immediately ...
111 views

Taylor series of a vector field?

Can someone give me the definition of the Taylor Series of a vector field in $\mathbb{C}^2$? Thanks!
729 views

2k views

Vector taylor series

Classical Electrodynamics by Jackson says "With a Taylor series expansion of the well-behaved $\rho (\mathbf{x'})$ around $\mathbf{x'} = \mathbf{x}$ one finds ..." and then he says basically that ...
83 views

When is the limit in $y$ of a Taylor expansion in $x$ a valid expansion?

I'd be interested to know when, if $$f(x,y)=g(x,y)+O(x^n)$$ we have that $$\lim_{y\rightarrow c}=\lim_{y\rightarrow c}g(x,y)+O(x^n).$$ Are there conditions of $f$ and/or $g$ that make sure that this ...
105 views

Taylor's formula

Taylor's Formula Write taylor's formula for $F(x,y)= \sin(x)\sin(y)$ using $a=0$, $b=0$, and $n=2$. \sin(h)\sin(k)=hk−\frac 16h(h^2+3k^2)\cos\theta h\sin\theta k−\frac 16 k(3h^2+k^2)\sin\theta ...